Multiple equilibrium model

Multiple equilibrium model, 24 131 Multiple extraction procedure (MEP), 25 869... 

The thermodynamics of micelle formation has been studied extensively. There is for example a mass action model (Wennestrdm and Lindman, 1979) that assumes that micelles can be described by an aggregate Mm with a single aggregation number m, so that the only descriptive equation is mMi Mm. A more complex form assumes the multiple equilibrium model, allowing aggregates of different sizes to be in equilibrium with each other (Tanford, 1978 Wennestrdm and Lindman, 1979 Israelachvili, 1992). 

In most cases when the micelle size distribution has been analyzed quantitatively the multiple equilibrium model has been used. This model can be formulated either, in analogy with Eq. (3.2), as a number of equilibria... 

The multiple-equilibrium model which corresponds to the scheme of Eq. (3) does not provide any critical concentration and is considered in the frame of the present discussion to be the opposite limiting case with regard to those detergent systems which had to be assigned to the phase separation model. 

Summarizing the statements of these three most commonly used models, it appears that the so-called mass action and phase-separation models simulate a third condition which must be fulfilled with respect to the formation of micelles a size limiting process. The latter is independent of the cooperativity and has to be interpreted by a molecular model. The limitation of the aggregate size in the mass action model is determined by the aggregation number. This is, essentially, the reason that this model has been preferred in the description of micelle forming systems. The multiple equilibrium model as comprised by the Eqs. (10—13) contains no such size limiting features. An improvement in this respect requires a functional relationship between the equilibrium constants and the association number n, i.e.,... 

The micelle has too small an aggregation number to be considered as a phase in the usual sense, and yet normally contains too many surfactant molecules to be considered as a chemical species. It is this dichotomy that makes an exact theory of solubilization by micelles difficult. The primary theoretical approaches to the problem are based on either a pseudophase model, mass action model, multiple equilibrium model, or the thermodynamics of small systems [191-196]. Technically, bulk thermodynamics should not apply to solute partitioning into small aggregates, since these solvents are interfacial phases with large surface-to-volume ratios. In contrast to a bulk phase, whose properties are invariant with position, the properties of small aggregates are expected to vary with distance from the interface [195]. The lattice model of solute partitioning concludes that virtually all types of solutes should favor the interface over the interior of a spherical micelle. While for cylindrical micelles, the internal distribution of solutes... 

Although the mass action approach could account for a number of experimental results, such as the small change in properties around the c.m.c., it has not escaped criticism. For example, the assumption that surfactants exist in solution in only two forms, namely single ions and micelles of uniform size, is debatable. Analysis of various experimental results has shown that micelles have a size distribution that is narrow and concentration dependent. Thus, the assumption of a single aggregation number is an oversimplification and, in reaUty, there is a micellar size distribution. This can be analyzed using the multiple equilibrium model, which can be best formulated as a stepwise aggregation [2],... 

An essentially equivalent approach to that of small-systems thermodynamics has been formulated by Corkill and co-workers and applied to systems of nonionic surfactants [94,176]. As with the small-systems approach, this multiple-equilibrium model considers equilibria between all micellar species present in solution rather than a single micellar species, as was considered by the mass-action theory. The intrinsic properties of the individual micellar species are then removed from the relationships by a suitable averaging procedure. The standard free energy and enthalpy of micellization are given by equations of similar form to Equations 3.44 and 3.45 and are shown to approximate satisfactorily to the appropriate mass-action equations for systems in which the mean aggregation number exceeds 20. 

Application of the small-systems/multiple-equilibrium models to solutions of ionic surfactants is less satisfactory because of the failure of these models to deal with interactions between micelles or to give a satisfactory description of the role of counterions in micellization. Alternative, more rigorous, thermodynamic formalisms for describing systems of interacting aggregates have been developed by Hall [184-188]. This theoretical approach has led to precise expressions for the effect of temperature, pressure and electrolyte concentration on the CMC which allow for solution non-ideality. Although these expressions... 

As the multiple-equilibrium model (Equation 1.20) contains a large number of equilibrium constants, drastic simplifying assumptions about the relations between them must be made in order to derive a relationship between experimentally determined relaxation times and rate constants of micelle formation as expressed by Equation 1.20. Kresheck et al. assumed that the rate-determining step is the loss of the first monomer from the micelle. In other words, the micelle reluctantly parts with one monomer molecule and then explodes. The role of equilibria involving the associations of intermediate species such that... 

A discussion of the different types of assumption that can be made in two-phase flow models is given in Chapter 9. DIERS[8] recommended the use of the homogeneous equilibrium model (HEM) for relief sizing, and so, preferably, a code which implements the HEM should be chosen. The model will need to incorporate sufficiently non-ideal modelling of physical properties and provision for multiple line diameters and potential choke points, as required by the application. 

Another very important feature of the stochastic equations considered here, when they are subjected to RMT analysis, is their resemblance to the general formalism arrived at in the thermodynamics of nonequilibrium processes this suggests an analogy between the effects of multiplicative noise and the continuous flux of energy which maintains the systems far from equilibrium. This is considered the main characteristic of self-organizing living systems and means that multiplicative stochastic models could take on a new and fundamentally important role. 

The hierarchical patch dynamic paradigm (HPDP) meets the above requirements (Wu and Loucks 1995, Wu and David 2002). It is a model for describing at a fundamental level the interactions and dynamics of ecological systems at landscape and regional scales. The HPDP inherently incorporates and predicts a variety of temporal and spatial scales, heterogeneity, and a wide range of dynamics. The basic tenets of HPDP are listed in Table 2.1. This framework is an alternative to models of ecological systems that incorporate a balance of nature, inherent stability, or multiple equilibriums. 

The most inclusive model to account for the nonlinear dependence of biological activity on logP is that derived by Martin and Hackbarth (162). It is an equilibrium model based on partition between multiple aqueous and non-aqueous compartments, similar to the one presented by Hyde (158). The inclusion of ionization processes of acids and conjugated acids (e.g., protonated amines) in the model and the analysis of the effect of these processes on the partition coefficient and the interactions with receptors makes this model generally applicable. 

Other workers [112, 113] have shown that a chemical equilibrium model of hydrocarbons based on an exponential-6 fluid model using Ross s soft-sphere perturbation theory is successful in reproducing the behavior of shocked hydrocarbons. Our model of the supercritical phase includes the species H2, CH4, C2H6, and C2H4. We have chosen model parameters to match both static compression isotherms and shock measurements wherever possible. The ability to match multiple types of experiments well increases confidence in the general applicability of our high-pressure equation of state model. 

Chang and Rochelle (16) developed an enhancement factor model based on approximate surface renewal with multiple equilibrium reactions. Their model included equilibria among and diffusion of the solution species H"1", SO2, HSO3, SO3, H2A,... 

Unfortunately, few of the published studies of extraction equilibria heve provided complete quantitative models that are useful for extrapolation of data or for predicting multiple metal distribution equilibria from single metal data. The chemical-reaction equilibrium formulation provides a framework for constructing such models. One of the drawbacks of purely empirical correlations of distribution coefficients is that pH has often been chosen as an independent variable. Such a choice is suggested by the form of Pigs. 8-3-5 and 8.3-8. Although pH is readily measured and contmlled on a laboratory scale, it is really a dependent variable, which is detenmined by mass belances and simultaneous reaction equilibria. An appropriate phare-equilibrium model should be able to predict equilibrium pH, at least within a moderate activity coefficient correction, concurrently with other species concemrations. 

An interesting aspect of kinetics is tha question of how species interact when multiple extraction inactions occur. Most extractants are not perfectly selective and are capable of extracting several species from a multicomponent equeous solution. Complete equilibrium models should indicate extraction selectivity for a given system at equilibrium, but selectivity may vary considerably in a nonequilibrium process because of rate differences. In fact, iron(UI) can be extracted by some of the copper-chelating extractants, but their effective selectivity for copper is based on the very slow kinetics of iron extraction. 

Aggregation of surfactants in apolar solvents, e.g., aliphatic or aromatic hydrocarbons, occurs provided that small amounts of water are present [1,126,127], These aggregates are often called reverse micelles, although the solutions do not always appear to have a critical micelle concentration, and surfactant association is often governed by a multiple equilibrium, mass action, model vith a large spread of aggregate sizes [130,131], It has recently been suggested that the existence of a monomer f -mer equilibrium should be used as a criterion of micellization, and that this term should not be applied to self-associated systems which involve multiple equilibria [132],... 

This is the general equation for multiple equilibrium (X and 11 are respective symbols for summation and multiplication) that describes the binding of ligand molecules to the biomacromolecules with multiple sites. However, there are so many adjustable parameters in this general equation, which is more revealing about the binding mechanism if the experimental data can be fitted to simplified models by more restrictive equations with fewer adjustable parameters. These models will be considered. 

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